The arithmetic mirror symmetry and Calabi-Yau manifolds

We extend our variant of mirror symmetry for K3 surfaces [GN3] and clarify its relation with mirror symmetry for Calabi-Yau manifolds. We introduce tw o classes (for the models A and B) of Calabi-Yau manifolds fibrated by K3 s urfaces with some special Picard lattices. These two classes are related wi th automorphic forms on IV type domains which we studied in our papers [GN1 ]-[GN6]. Conjecturally these automorphic forms take part in the quantum int ersection pairing for model A, Yukawa coupling for model B and mirror symme try between these two classes of Calabi-Yau manifolds. Recently there were several papers by physicists where it was shown on some examples. We propos e a problem of classification of introduced Calabi-Yau manifolds. Our paper s [GN1]-[GN6] and [N3]-[N14] give hope that this is possible. They describe possible Picard or transcendental lattices of general K3 fibers of the Cal abi-Yau manifolds.